A factor is a number that divides evenly into another number, with no remainder. This page covers factors and multiples, prime factorization with factor trees, greatest common factor and least common multiple, and a first look at factoring in algebra.
What are factors and multiples?
A factor is a number or expression that divides another number evenly, leaving no remainder (Britannica). A multiple is what you get when you multiply a number by a whole number.
The factors of 12 are 1, 2, 3, 4, 6 and 12, because each one divides 12 evenly. The multiples of 4 are 4, 8, 12, 16, 20 and so on, since each one is 4 times a whole number.
To find all the factors of a number, test each whole number starting at 1 to see if it divides evenly. For 18: 1 x 18, 2 x 9, and 3 x 6 all work, so the factors of 18 are 1, 2, 3, 6, 9 and 18.
This section covers whole-number factors and multiples. It doesn’t cover factoring negative numbers or fractions, which follow related but separate rules.
What is prime factorization, and how do factor trees work?
A prime number is a whole number greater than 1 that’s divisible only by 1 and itself (Britannica). The numbers 2, 3, 5, 7, 11 and 13 are prime. A number with more than two factors, like 12, is called composite. Prime factorization means breaking a number down into the prime numbers that multiply together to make it.
A factor tree is a diagram that helps with this. You split a number into any two factors, then keep splitting each factor that isn’t prime yet, until every branch ends in a prime number.
Worked example. Find the prime factorization of 36.
- Split 36 into 6 x 6.
- Neither 6 is prime, so split each one: 6 = 2 x 3, and the other 6 = 2 x 3.
- All four branches now end in primes: 2, 3, 2, 3.
So the prime factorization of 36 is 2 x 2 x 3 x 3, usually written 2² x 3².
Another example. Find the prime factorization of 60. Split 60 into 6 x 10. Split 6 into 2 x 3. Split 10 into 2 x 5. Every branch is now prime: 2, 3, 2, 5. The prime factorization of 60 is 2 x 2 x 3 x 5, or 2² x 3 x 5.
This section covers prime factorization of whole numbers under a few hundred. It doesn’t cover checking whether very large numbers are prime, which needs faster methods than a factor tree.
How do you find the GCF and LCM?
The greatest common factor (GCF) of two numbers is the largest factor they share. The least common multiple (LCM) is the smallest multiple they share.
Worked example, GCF. Find the GCF of 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The factors they share are 1, 2, 3 and 6. The greatest of those is 6, so the GCF of 12 and 18 is 6.
Worked example, LCM. Find the LCM of 6 and 9. Multiples of 6: 6, 12, 18, 24, 30. Multiples of 9: 9, 18, 27, 36. The smallest multiple they share is 18, so the LCM of 6 and 9 is 18.
The GCF is what you use to simplify a fraction, and the LCM is what you use to find a common denominator when adding or subtracting fractions, covered on our fractions page.
This section covers finding GCF and LCM by listing factors and multiples. It doesn’t cover finding them from prime factorizations, which is faster for larger numbers but is a separate technique.
How does factoring work in algebra?
Factoring an algebraic expression means rewriting it as a product of simpler expressions, the reverse of multiplying them out. The simplest kind pulls out a common factor shared by every term.
Worked example, common factor. Factor 6x + 9. Both terms share a common factor of 3: 6x = 3 x 2x, and 9 = 3 x 3. Pull out the 3: 6x + 9 = 3(2x + 3).
Checking it. Multiply 3(2x + 3) back out: 3 x 2x = 6x, and 3 x 3 = 9, giving 6x + 9 again, which matches the original.
This section covers factoring out a common factor from a simple expression. It doesn’t cover factoring trinomials like x² + 5x + 6, which needs finding two numbers that multiply to the last term and add to the middle term, a technique for a later algebra lesson.
Practice set
Work through these on paper, then check your answers.
- List all the factors of 24.
- Find the prime factorization of 48.
- Find the GCF of 15 and 20.
- Find the LCM of 9 and 21.
- Factor 8x + 12 using a common factor.
Show answers and reasoning
- The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
- 48 = 2 x 24 = 2 x 2 x 12 = 2 x 2 x 2 x 6 = 2 x 2 x 2 x 2 x 3, or 2⁴ x 3.
- The factors of 15 are 1, 3, 5, 15. The factors of 20 are 1, 2, 4, 5, 10, 20. The greatest shared factor is 5, so the GCF is 5.
- Multiples of 9: 9, 18, 27, 36, 45, 54, 63. Multiples of 21: 21, 42, 63. The smallest shared multiple is 63, so the LCM is 63.
- Both terms share a common factor of 4: 8x = 4 x 2x, and 12 = 4 x 3. So 8x + 12 = 4(2x + 3). Check: 4 x 2x = 8x, and 4 x 3 = 12.
Factoring is the skill behind simplifying and combining fractions, and it comes back constantly once algebra moves into equations and expressions. It also connects to squares and square roots, since a perfect square is really just a number with a repeated factor. For more topics, visit our math hub.